Plitt Modified Hydrocyclone Alpha

Summary

The Plitt Modified Hydrocyclone Alpha is intended to provide a modified Plitt-family option to represents hydrocyclone classification using a modified empirical Plitt-family formulation. The model calculates pressure, volumetric split, corrected cut size, sharpness of separation, water bypass and the size-by-size partition to underflow.

DPSIM model key: DPSIM.Classification.PlittModifiedHydrocycloneAlpha
Category: Classification
Subcategory: Hydrocyclones
Display name: Plitt Modified Hydrocyclone Alpha

Parameters

Derived parameters

Model Description

The Plitt Modified Hydrocyclone Alpha model receives one feed stream and generates two product streams. In DPSIM, the product port represents the underflow and the tail port represents the overflow.

The total feed volumetric flowrate is divided by the number of hydrocyclones in parallel:

Q_i=Q_F/N_parallel

Where:

The model uses the feed volumetric solids fraction:

φ_v=V_S^F/V_P^F

Where:

The implemented head-like pressure correlation is:

H=A_1 Q_i^1.46 exp(-7.63φ_v+10.79φ_v^2)/(D_c^0.20 h^0.15 D_i^0.51 D_o^1.65 D_u^0.53)

The pressure is then calculated as:

P=H ρ_p 0.433 6.895 F_H

Where:

Difference from Plitt Original: Alpha does not use the original metric pressure equation directly. It uses an alternative fitted head/pressure correlation with geometry entered in inches and flowrate in m3/h per cyclone.

The base volumetric split ratio is calculated as:

S=A_3 h^0.19 (D_u/D_o)^2.64 exp(-4.33φ_v+8.77φ_v^2)/(H^0.54 D_c^0.38)

Where:

Difference from Plitt Original: the original Plitt split equation uses a different coefficient, includes (D_u^2+D_o^2)^0.36, and uses an exponential term based on percent solids by volume. Alpha uses a different fitted split equation with a quadratic solids-concentration term.

For each component c, the component split is:

S_c=S F_(S,c)

Where:

The sharpness parameter is calculated as:

m=exp(A_4-1.58 S_c/(S_c+1))(D_c^2 h/Q_i)^0.15

The component-adjusted sharpness is:

m_c=m F_(m,c)

Where:

Difference from Plitt Original: Plitt relates sharpness to the volumetric recovery to underflow and the geometric term D_c^2 h/Q. Alpha keeps the same general dependence but uses the modified Alpha coefficient A4 and the component-adjusted split.

The corrected cut size is calculated as:

d_(50,c)=A_2 D_c^0.44 D_i^0.58 D_o^1.91 exp(11.12φ_v)/(D_u^0.80 h^0.37 Q_i^0.44 (ρ_(s,c)-1)^0.5)

The calibrated component cut size is:

d_(50,c)^*=d_(50,c) F_(d50,c)

Where:

Difference from Plitt Original: the Alpha cut-size equation does not use the original Plitt exponents and constant. The dependence on D_o is stronger, the density term is simplified as (ρ_s-1)^0.5, and the solids-concentration correction is exp(11.12φ_v) rather than the original metric expression.

The corrected Rosin-Rammler partition curve is:

Y'(c,i)=1-exp(-0.693(d_i/d(50,c)^*)^m_c)

Where:

The model calculates a hypothetical solids recovery from the feed PSD and the corrected curve:

R_sc=sum_i f_i Y'_i

Where:

The water bypass to underflow is calculated as:

B_pw=(S_c/(S_c+1)-φ_v R_sc)/(1-φ_v(1-λ(1-R_sc)))

The feed bypass term is:

R_f=λ B_pw

Both B_pw and R_f are limited to the range from 0 to 1.

Where:

Difference from Plitt Original: Plitt Original calculates liquid recovery to underflow through a relationship involving volumetric recovery and solids recovery. Alpha uses a modified lambda-adjusted bypass expression and does not solve a full iterative Plitt liquid-recovery calculation.

The coarse bypass is limited internally:

B_pc^*=min(B_pc,1-R_f)

The final partition to underflow is:

E_(c,i)=R_f+(1-R_f-B_pc^*)Y'_(c,i)

Where:

Difference from Plitt Original: the explicit coarse bypass term B_pc is a DPSIM/Alpha extension. It is not part of the original Plitt corrected partition formulation.

The underflow and overflow component-size masses are calculated as:

M_(U,c,i)=E_(c,i) M_(F,c,i)

M_(O,c,i)=M_(F,c,i)-M_(U,c,i)

The total retained size distributions are calculated by summing over components:

M_(U,i)=sum_c M_(U,c,i)

M_(O,i)=sum_c M_(O,c,i)

p_(U,i)=M_(U,i)/sum_j M_(U,j)

p_(O,i)=M_(O,i)/sum_j M_(O,j)

The component fractions in each size interval are:

z_(U,c,i)=M_(U,c,i)/M_(U,i)

z_(O,c,i)=M_(O,c,i)/M_(O,i)

Where:

The water split is calculated from the base hydraulic water bypass:

W_U=B_pw,base W_F

W_O=W_F-W_U

Where:

The product stream corresponds to the underflow. The tail stream corresponds to the overflow.

The model preserves total solids by splitting each component-size mass between underflow and overflow. Water is split using the base hydraulic water bypass.

The parameters d50 factor, imperfection factor and split factor are component-specific calibration factors.

The pressure-control extension is not part of the original Plitt model.

When pressure control is enabled, the model compares the calculated pressure with the high and low pressure thresholds. If pressure is above the high threshold, the number of operating hydrocyclones is increased by one, limited by the maximum number available. If pressure is below the low threshold, the number of operating hydrocyclones is decreased by one, limited by the minimum number available.

The hydraulic constants A1, A2, A3 and A4 are empirical and must be calibrated before design use.

References

Plitt, L. R. (1976). A Mathematical Model of the Hydrocyclone Classifier. CIM Bulletin, December 1976, 114–123.